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On the characterization of weighted simple games

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Abstract

This paper has a twofold scope. The first one is to clarify and put in evidence the isomorphic character of two theories developed in quite different fields: on one side, threshold logic, on the other side, simple games. One of the main purposes in both theories is to determine when a simple game is representable as a weighted game, which allows a very compact and easily comprehensible representation. Deep results were found in threshold logic in the sixties and seventies for this problem. However, game theory has taken the lead and some new results have been obtained for the problem in the past two decades. The second and main goal of this paper is to provide some new results on this problem and propose several open questions and conjectures for future research. The results we obtain depend on two significant parameters of the game: the number of types of equivalent players and the number of types of shift-minimal winning coalitions.

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Notes

  1. The minimal winning coalitions are given by \(\{A,C\}\), \(\{A,D\}\), \(\{B,C\}\), and \(\{B,D\}\), see Sect. 2 for the definitions.

  2. Using the notation from Sect. 3, \(\langle \{A,C\},\{B,D\}\,| \,\{A,B\},\{C,D\}\rangle \) is a trading transform, which certifies non-weightedness.

  3. There is no connection to the efficient computation of power indices. In general, generating functions are just a theoretical tool from enumerative combinatorics in order to compute exact formulas for recurrence relations.

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Acknowledgements

This research was partially supported by funds from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER Funds) under Grant MTM2015-66818-P (MINECO/FEDER).

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Authors and Affiliations

  1. Department of Mathematics, Universitat Politècnica de Catalunya (Campus Manresa), 08242, Manresa, Spain

    Josep Freixas

  2. Cirprotec, 08233, Terrassa, Spain

    Marc Freixas

  3. Department of Mathematics, University of Bayreuth, 95440, Bayreuth, Germany

    Sascha Kurz

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  1. Josep Freixas
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Freixas, J., Freixas, M. & Kurz, S. On the characterization of weighted simple games. Theory Decis 83, 469–498 (2017). https://doi.org/10.1007/s11238-017-9606-z

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